YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
f(a(),b()) -> c()
a() -> a'()
b() -> b'()
c() -> f(a'(),b())
c() -> f(a(),b'())
c() -> f(a(),b())

CP(R):
f(a'(),b()) -> c()
f(a(),b'()) -> c()
f(a'(),b()) -> f(a(),b'())
f(a'(),b()) -> f(a(),b())
f(a(),b'()) -> f(a'(),b())
f(a(),b'()) -> f(a(),b())
f(a(),b()) -> f(a'(),b())
f(a(),b()) -> f(a(),b'())

critical-pair-closing system C:
a() -> a'()
b() -> b'()
c() -> f(a(),b'())

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
a() -> a'()
b() -> b'()
c() -> f(a(),b'())
CP(R):